arXiv research

Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.

GCNs converge and remain stable on large random graphs, revealing geometric insights.

The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider $\varepsilon$-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of $\mathbb{R}^K$. We show t…

In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erdös-Renyi random graphs, geometric random graphs, expanders and random $k$-regular graphs. Moreover we show that …

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the …

Graph Neural Networks struggle on random graphs without node identifiers.

Estimating dimension from sparse random geometric graphs.

Algorithm reconstructs vertex positions in random geometric graphs with improved accuracy.

Graph kernels are widely used for measuring the similarity between graphs. Many existing graph kernels, which focus on local patterns within graphs rather than their global properties, suffer from significant structure information loss when representing graphs. Some recent global graph kernels, which utilizes the align…

This paper tackles matching two complete graphs with correlated edge weights in geometric models.

New methods learn from single graphs, improving transductive node classification.

Graph matching with feature vectors is solved using a two-layer graph neural network.

GATs improve node regression on noisy graphs with provable advantage.

Estimates latent norms and Gram matrices for graphs on Euclidean balls.

A graph clustering method that moves nodes to highest-degree neighbors.

Study shows how discrete graph curvature relates to manifold curvature.

New method beats volumetric barrier for manifold recovery.

New method calculates Ricci curvature from distances between weighted volumes.

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of…

We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the…

Researchers identify critical protein residues using advanced graph theory.

Large unweighted directed graphs are commonly used to capture relations between entities. A fundamental problem in the analysis of such networks is to properly define the similarity or dissimilarity between any two vertices. Despite the significance of this problem, statistical characterization of the proposed metrics …

FGN models networks with fractal structures using Gaussian Multiplicative Chaos.

IDPGs extend RDPGs with a Poisson process for random latent positions.

New model calculates logarithmic surface diameter.

We study two global structural properties of a graph $Γ$, denoted AS and CFS, which arise in a natural way from geometric group theory. We study these properties in the Erdös--Rényi random graph model G(n,p), proving a sharp threshold for a random graph to have the AS property asymptotically almost surely, and giving f…

The study extends stochastic block models to geometric settings, focusing on community detection and information flow.

Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.

Bayesian models for networks are often misspecified, leading to overconfident inference.

Random covers of hyperbolic surfaces follow a specific probability measure.

A new growth model for dynamic networks using Markovian latent points.

New method recovers graph latent positions under edge differential privacy.

The paper proves Lipschitz regularity of graph Laplacian eigenvectors on random data clouds.

Study reveals limits of detecting local geometry in random graphs.

Random hyperbolic surfaces are mostly tangle-free, with geometric implications.

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

The paper tracks patient recovery using graphs of joint movement data.

Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a ``typical'' compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; b…

The article studies random infinite ideal hyperbolic polyhedra and their dual graphs, establishing new boundary theories.

The paper predicts responses on out-of-sample nodes using latent positions on unknown curves.

Study spectral properties of sparse random graphs to recover latent vectors.

A new graph-based clustering method for moderate-dimensional data.

A parametrization of hypergraphs based on the geometry of points in $\mathbf{R}^d$ is developed. Informative prior distributions on hypergraphs are induced through this parametrization by priors on point configurations via spatial processes. This prior specification is used to infer conditional independence models or M…

Study matches two noisy point clouds with geometric transformations and relabeling.

The geometric block model is a recently proposed generative model for random graphs that is able to capture the inherent geometric properties of many community detection problems, providing more accurate characterizations of practical community structures compared with the popular stochastic block model. Galhotra et al…

The problem of completing high-dimensional matrices from a limited set of observations arises in many big data applications, especially, recommender systems. Existing matrix completion models generally follow either a memory- or a model-based approach, whereas, geometric matrix completion models combine the best from b…

The paper bounds crossing numbers of dense graphs on surfaces.

Study on communication delays in decentralized learning networks.